# HG changeset patch
# User huffman
# Date 1268069585 28800
# Node ID f87132febfac789dc0d5f6bbb931e1150dc493d4
# Parent  05ca920cd94b23fd7eb50d2452e5c27a4066c02b
move lemmas from Domain.thy to Domain_Aux.thy

diff -r 05ca920cd94b -r f87132febfac src/HOLCF/Domain.thy
--- a/src/HOLCF/Domain.thy	Mon Mar 08 08:12:48 2010 -0800
+++ b/src/HOLCF/Domain.thy	Mon Mar 08 09:33:05 2010 -0800
@@ -19,107 +19,6 @@
 defaultsort pcpo
 
 
-subsection {* Continuous isomorphisms *}
-
-text {* A locale for continuous isomorphisms *}
-
-locale iso =
-  fixes abs :: "'a \<rightarrow> 'b"
-  fixes rep :: "'b \<rightarrow> 'a"
-  assumes abs_iso [simp]: "rep\<cdot>(abs\<cdot>x) = x"
-  assumes rep_iso [simp]: "abs\<cdot>(rep\<cdot>y) = y"
-begin
-
-lemma swap: "iso rep abs"
-  by (rule iso.intro [OF rep_iso abs_iso])
-
-lemma abs_below: "(abs\<cdot>x \<sqsubseteq> abs\<cdot>y) = (x \<sqsubseteq> y)"
-proof
-  assume "abs\<cdot>x \<sqsubseteq> abs\<cdot>y"
-  then have "rep\<cdot>(abs\<cdot>x) \<sqsubseteq> rep\<cdot>(abs\<cdot>y)" by (rule monofun_cfun_arg)
-  then show "x \<sqsubseteq> y" by simp
-next
-  assume "x \<sqsubseteq> y"
-  then show "abs\<cdot>x \<sqsubseteq> abs\<cdot>y" by (rule monofun_cfun_arg)
-qed
-
-lemma rep_below: "(rep\<cdot>x \<sqsubseteq> rep\<cdot>y) = (x \<sqsubseteq> y)"
-  by (rule iso.abs_below [OF swap])
-
-lemma abs_eq: "(abs\<cdot>x = abs\<cdot>y) = (x = y)"
-  by (simp add: po_eq_conv abs_below)
-
-lemma rep_eq: "(rep\<cdot>x = rep\<cdot>y) = (x = y)"
-  by (rule iso.abs_eq [OF swap])
-
-lemma abs_strict: "abs\<cdot>\<bottom> = \<bottom>"
-proof -
-  have "\<bottom> \<sqsubseteq> rep\<cdot>\<bottom>" ..
-  then have "abs\<cdot>\<bottom> \<sqsubseteq> abs\<cdot>(rep\<cdot>\<bottom>)" by (rule monofun_cfun_arg)
-  then have "abs\<cdot>\<bottom> \<sqsubseteq> \<bottom>" by simp
-  then show ?thesis by (rule UU_I)
-qed
-
-lemma rep_strict: "rep\<cdot>\<bottom> = \<bottom>"
-  by (rule iso.abs_strict [OF swap])
-
-lemma abs_defin': "abs\<cdot>x = \<bottom> \<Longrightarrow> x = \<bottom>"
-proof -
-  have "x = rep\<cdot>(abs\<cdot>x)" by simp
-  also assume "abs\<cdot>x = \<bottom>"
-  also note rep_strict
-  finally show "x = \<bottom>" .
-qed
-
-lemma rep_defin': "rep\<cdot>z = \<bottom> \<Longrightarrow> z = \<bottom>"
-  by (rule iso.abs_defin' [OF swap])
-
-lemma abs_defined: "z \<noteq> \<bottom> \<Longrightarrow> abs\<cdot>z \<noteq> \<bottom>"
-  by (erule contrapos_nn, erule abs_defin')
-
-lemma rep_defined: "z \<noteq> \<bottom> \<Longrightarrow> rep\<cdot>z \<noteq> \<bottom>"
-  by (rule iso.abs_defined [OF iso.swap]) (rule iso_axioms)
-
-lemma abs_defined_iff: "(abs\<cdot>x = \<bottom>) = (x = \<bottom>)"
-  by (auto elim: abs_defin' intro: abs_strict)
-
-lemma rep_defined_iff: "(rep\<cdot>x = \<bottom>) = (x = \<bottom>)"
-  by (rule iso.abs_defined_iff [OF iso.swap]) (rule iso_axioms)
-
-lemma casedist_rule: "rep\<cdot>x = \<bottom> \<or> P \<Longrightarrow> x = \<bottom> \<or> P"
-  by (simp add: rep_defined_iff)
-
-lemma compact_abs_rev: "compact (abs\<cdot>x) \<Longrightarrow> compact x"
-proof (unfold compact_def)
-  assume "adm (\<lambda>y. \<not> abs\<cdot>x \<sqsubseteq> y)"
-  with cont_Rep_CFun2
-  have "adm (\<lambda>y. \<not> abs\<cdot>x \<sqsubseteq> abs\<cdot>y)" by (rule adm_subst)
-  then show "adm (\<lambda>y. \<not> x \<sqsubseteq> y)" using abs_below by simp
-qed
-
-lemma compact_rep_rev: "compact (rep\<cdot>x) \<Longrightarrow> compact x"
-  by (rule iso.compact_abs_rev [OF iso.swap]) (rule iso_axioms)
-
-lemma compact_abs: "compact x \<Longrightarrow> compact (abs\<cdot>x)"
-  by (rule compact_rep_rev) simp
-
-lemma compact_rep: "compact x \<Longrightarrow> compact (rep\<cdot>x)"
-  by (rule iso.compact_abs [OF iso.swap]) (rule iso_axioms)
-
-lemma iso_swap: "(x = abs\<cdot>y) = (rep\<cdot>x = y)"
-proof
-  assume "x = abs\<cdot>y"
-  then have "rep\<cdot>x = rep\<cdot>(abs\<cdot>y)" by simp
-  then show "rep\<cdot>x = y" by simp
-next
-  assume "rep\<cdot>x = y"
-  then have "abs\<cdot>(rep\<cdot>x) = abs\<cdot>y" by simp
-  then show "x = abs\<cdot>y" by simp
-qed
-
-end
-
-
 subsection {* Casedist *}
 
 lemma ex_one_defined_iff:
@@ -214,102 +113,6 @@
   ssum_map_sinl ssum_map_sinr sprod_map_spair u_map_up
 
 
-subsection {* Take functions and finiteness *}
-
-lemma lub_ID_take_lemma:
-  assumes "chain t" and "(\<Squnion>n. t n) = ID"
-  assumes "\<And>n. t n\<cdot>x = t n\<cdot>y" shows "x = y"
-proof -
-  have "(\<Squnion>n. t n\<cdot>x) = (\<Squnion>n. t n\<cdot>y)"
-    using assms(3) by simp
-  then have "(\<Squnion>n. t n)\<cdot>x = (\<Squnion>n. t n)\<cdot>y"
-    using assms(1) by (simp add: lub_distribs)
-  then show "x = y"
-    using assms(2) by simp
-qed
-
-lemma lub_ID_reach:
-  assumes "chain t" and "(\<Squnion>n. t n) = ID"
-  shows "(\<Squnion>n. t n\<cdot>x) = x"
-using assms by (simp add: lub_distribs)
-
-text {*
-  Let a ``decisive'' function be a deflation that maps every input to
-  either itself or bottom.  Then if a domain's take functions are all
-  decisive, then all values in the domain are finite.
-*}
-
-definition
-  decisive :: "('a::pcpo \<rightarrow> 'a) \<Rightarrow> bool"
-where
-  "decisive d \<longleftrightarrow> (\<forall>x. d\<cdot>x = x \<or> d\<cdot>x = \<bottom>)"
-
-lemma decisiveI: "(\<And>x. d\<cdot>x = x \<or> d\<cdot>x = \<bottom>) \<Longrightarrow> decisive d"
-  unfolding decisive_def by simp
-
-lemma decisive_cases:
-  assumes "decisive d" obtains "d\<cdot>x = x" | "d\<cdot>x = \<bottom>"
-using assms unfolding decisive_def by auto
-
-lemma decisive_bottom: "decisive \<bottom>"
-  unfolding decisive_def by simp
-
-lemma decisive_ID: "decisive ID"
-  unfolding decisive_def by simp
-
-lemma decisive_ssum_map:
-  assumes f: "decisive f"
-  assumes g: "decisive g"
-  shows "decisive (ssum_map\<cdot>f\<cdot>g)"
-apply (rule decisiveI, rename_tac s)
-apply (case_tac s, simp_all)
-apply (rule_tac x=x in decisive_cases [OF f], simp_all)
-apply (rule_tac x=y in decisive_cases [OF g], simp_all)
-done
-
-lemma decisive_sprod_map:
-  assumes f: "decisive f"
-  assumes g: "decisive g"
-  shows "decisive (sprod_map\<cdot>f\<cdot>g)"
-apply (rule decisiveI, rename_tac s)
-apply (case_tac s, simp_all)
-apply (rule_tac x=x in decisive_cases [OF f], simp_all)
-apply (rule_tac x=y in decisive_cases [OF g], simp_all)
-done
-
-lemma decisive_abs_rep:
-  fixes abs rep
-  assumes iso: "iso abs rep"
-  assumes d: "decisive d"
-  shows "decisive (abs oo d oo rep)"
-apply (rule decisiveI)
-apply (rule_tac x="rep\<cdot>x" in decisive_cases [OF d])
-apply (simp add: iso.rep_iso [OF iso])
-apply (simp add: iso.abs_strict [OF iso])
-done
-
-lemma lub_ID_finite:
-  assumes chain: "chain d"
-  assumes lub: "(\<Squnion>n. d n) = ID"
-  assumes decisive: "\<And>n. decisive (d n)"
-  shows "\<exists>n. d n\<cdot>x = x"
-proof -
-  have 1: "chain (\<lambda>n. d n\<cdot>x)" using chain by simp
-  have 2: "(\<Squnion>n. d n\<cdot>x) = x" using chain lub by (rule lub_ID_reach)
-  have "\<forall>n. d n\<cdot>x = x \<or> d n\<cdot>x = \<bottom>"
-    using decisive unfolding decisive_def by simp
-  hence "range (\<lambda>n. d n\<cdot>x) \<subseteq> {x, \<bottom>}"
-    by auto
-  hence "finite (range (\<lambda>n. d n\<cdot>x))"
-    by (rule finite_subset, simp)
-  with 1 have "finite_chain (\<lambda>n. d n\<cdot>x)"
-    by (rule finite_range_imp_finch)
-  then have "\<exists>n. (\<Squnion>n. d n\<cdot>x) = d n\<cdot>x"
-    unfolding finite_chain_def by (auto simp add: maxinch_is_thelub)
-  with 2 show "\<exists>n. d n\<cdot>x = x" by (auto elim: sym)
-qed
-
-
 subsection {* Installing the domain package *}
 
 lemmas con_strict_rules =
diff -r 05ca920cd94b -r f87132febfac src/HOLCF/Domain_Aux.thy
--- a/src/HOLCF/Domain_Aux.thy	Mon Mar 08 08:12:48 2010 -0800
+++ b/src/HOLCF/Domain_Aux.thy	Mon Mar 08 09:33:05 2010 -0800
@@ -10,6 +10,107 @@
   ("Tools/Domain/domain_take_proofs.ML")
 begin
 
+subsection {* Continuous isomorphisms *}
+
+text {* A locale for continuous isomorphisms *}
+
+locale iso =
+  fixes abs :: "'a \<rightarrow> 'b"
+  fixes rep :: "'b \<rightarrow> 'a"
+  assumes abs_iso [simp]: "rep\<cdot>(abs\<cdot>x) = x"
+  assumes rep_iso [simp]: "abs\<cdot>(rep\<cdot>y) = y"
+begin
+
+lemma swap: "iso rep abs"
+  by (rule iso.intro [OF rep_iso abs_iso])
+
+lemma abs_below: "(abs\<cdot>x \<sqsubseteq> abs\<cdot>y) = (x \<sqsubseteq> y)"
+proof
+  assume "abs\<cdot>x \<sqsubseteq> abs\<cdot>y"
+  then have "rep\<cdot>(abs\<cdot>x) \<sqsubseteq> rep\<cdot>(abs\<cdot>y)" by (rule monofun_cfun_arg)
+  then show "x \<sqsubseteq> y" by simp
+next
+  assume "x \<sqsubseteq> y"
+  then show "abs\<cdot>x \<sqsubseteq> abs\<cdot>y" by (rule monofun_cfun_arg)
+qed
+
+lemma rep_below: "(rep\<cdot>x \<sqsubseteq> rep\<cdot>y) = (x \<sqsubseteq> y)"
+  by (rule iso.abs_below [OF swap])
+
+lemma abs_eq: "(abs\<cdot>x = abs\<cdot>y) = (x = y)"
+  by (simp add: po_eq_conv abs_below)
+
+lemma rep_eq: "(rep\<cdot>x = rep\<cdot>y) = (x = y)"
+  by (rule iso.abs_eq [OF swap])
+
+lemma abs_strict: "abs\<cdot>\<bottom> = \<bottom>"
+proof -
+  have "\<bottom> \<sqsubseteq> rep\<cdot>\<bottom>" ..
+  then have "abs\<cdot>\<bottom> \<sqsubseteq> abs\<cdot>(rep\<cdot>\<bottom>)" by (rule monofun_cfun_arg)
+  then have "abs\<cdot>\<bottom> \<sqsubseteq> \<bottom>" by simp
+  then show ?thesis by (rule UU_I)
+qed
+
+lemma rep_strict: "rep\<cdot>\<bottom> = \<bottom>"
+  by (rule iso.abs_strict [OF swap])
+
+lemma abs_defin': "abs\<cdot>x = \<bottom> \<Longrightarrow> x = \<bottom>"
+proof -
+  have "x = rep\<cdot>(abs\<cdot>x)" by simp
+  also assume "abs\<cdot>x = \<bottom>"
+  also note rep_strict
+  finally show "x = \<bottom>" .
+qed
+
+lemma rep_defin': "rep\<cdot>z = \<bottom> \<Longrightarrow> z = \<bottom>"
+  by (rule iso.abs_defin' [OF swap])
+
+lemma abs_defined: "z \<noteq> \<bottom> \<Longrightarrow> abs\<cdot>z \<noteq> \<bottom>"
+  by (erule contrapos_nn, erule abs_defin')
+
+lemma rep_defined: "z \<noteq> \<bottom> \<Longrightarrow> rep\<cdot>z \<noteq> \<bottom>"
+  by (rule iso.abs_defined [OF iso.swap]) (rule iso_axioms)
+
+lemma abs_defined_iff: "(abs\<cdot>x = \<bottom>) = (x = \<bottom>)"
+  by (auto elim: abs_defin' intro: abs_strict)
+
+lemma rep_defined_iff: "(rep\<cdot>x = \<bottom>) = (x = \<bottom>)"
+  by (rule iso.abs_defined_iff [OF iso.swap]) (rule iso_axioms)
+
+lemma casedist_rule: "rep\<cdot>x = \<bottom> \<or> P \<Longrightarrow> x = \<bottom> \<or> P"
+  by (simp add: rep_defined_iff)
+
+lemma compact_abs_rev: "compact (abs\<cdot>x) \<Longrightarrow> compact x"
+proof (unfold compact_def)
+  assume "adm (\<lambda>y. \<not> abs\<cdot>x \<sqsubseteq> y)"
+  with cont_Rep_CFun2
+  have "adm (\<lambda>y. \<not> abs\<cdot>x \<sqsubseteq> abs\<cdot>y)" by (rule adm_subst)
+  then show "adm (\<lambda>y. \<not> x \<sqsubseteq> y)" using abs_below by simp
+qed
+
+lemma compact_rep_rev: "compact (rep\<cdot>x) \<Longrightarrow> compact x"
+  by (rule iso.compact_abs_rev [OF iso.swap]) (rule iso_axioms)
+
+lemma compact_abs: "compact x \<Longrightarrow> compact (abs\<cdot>x)"
+  by (rule compact_rep_rev) simp
+
+lemma compact_rep: "compact x \<Longrightarrow> compact (rep\<cdot>x)"
+  by (rule iso.compact_abs [OF iso.swap]) (rule iso_axioms)
+
+lemma iso_swap: "(x = abs\<cdot>y) = (rep\<cdot>x = y)"
+proof
+  assume "x = abs\<cdot>y"
+  then have "rep\<cdot>x = rep\<cdot>(abs\<cdot>y)" by simp
+  then show "rep\<cdot>x = y" by simp
+next
+  assume "rep\<cdot>x = y"
+  then have "abs\<cdot>(rep\<cdot>x) = abs\<cdot>y" by simp
+  then show "x = abs\<cdot>y" by simp
+qed
+
+end
+
+
 subsection {* Proofs about take functions *}
 
 text {*
@@ -45,6 +146,104 @@
   ultimately show ?thesis by simp
 qed
 
+lemma lub_ID_take_lemma:
+  assumes "chain t" and "(\<Squnion>n. t n) = ID"
+  assumes "\<And>n. t n\<cdot>x = t n\<cdot>y" shows "x = y"
+proof -
+  have "(\<Squnion>n. t n\<cdot>x) = (\<Squnion>n. t n\<cdot>y)"
+    using assms(3) by simp
+  then have "(\<Squnion>n. t n)\<cdot>x = (\<Squnion>n. t n)\<cdot>y"
+    using assms(1) by (simp add: lub_distribs)
+  then show "x = y"
+    using assms(2) by simp
+qed
+
+lemma lub_ID_reach:
+  assumes "chain t" and "(\<Squnion>n. t n) = ID"
+  shows "(\<Squnion>n. t n\<cdot>x) = x"
+using assms by (simp add: lub_distribs)
+
+
+subsection {* Finiteness *}
+
+text {*
+  Let a ``decisive'' function be a deflation that maps every input to
+  either itself or bottom.  Then if a domain's take functions are all
+  decisive, then all values in the domain are finite.
+*}
+
+definition
+  decisive :: "('a::pcpo \<rightarrow> 'a) \<Rightarrow> bool"
+where
+  "decisive d \<longleftrightarrow> (\<forall>x. d\<cdot>x = x \<or> d\<cdot>x = \<bottom>)"
+
+lemma decisiveI: "(\<And>x. d\<cdot>x = x \<or> d\<cdot>x = \<bottom>) \<Longrightarrow> decisive d"
+  unfolding decisive_def by simp
+
+lemma decisive_cases:
+  assumes "decisive d" obtains "d\<cdot>x = x" | "d\<cdot>x = \<bottom>"
+using assms unfolding decisive_def by auto
+
+lemma decisive_bottom: "decisive \<bottom>"
+  unfolding decisive_def by simp
+
+lemma decisive_ID: "decisive ID"
+  unfolding decisive_def by simp
+
+lemma decisive_ssum_map:
+  assumes f: "decisive f"
+  assumes g: "decisive g"
+  shows "decisive (ssum_map\<cdot>f\<cdot>g)"
+apply (rule decisiveI, rename_tac s)
+apply (case_tac s, simp_all)
+apply (rule_tac x=x in decisive_cases [OF f], simp_all)
+apply (rule_tac x=y in decisive_cases [OF g], simp_all)
+done
+
+lemma decisive_sprod_map:
+  assumes f: "decisive f"
+  assumes g: "decisive g"
+  shows "decisive (sprod_map\<cdot>f\<cdot>g)"
+apply (rule decisiveI, rename_tac s)
+apply (case_tac s, simp_all)
+apply (rule_tac x=x in decisive_cases [OF f], simp_all)
+apply (rule_tac x=y in decisive_cases [OF g], simp_all)
+done
+
+lemma decisive_abs_rep:
+  fixes abs rep
+  assumes iso: "iso abs rep"
+  assumes d: "decisive d"
+  shows "decisive (abs oo d oo rep)"
+apply (rule decisiveI)
+apply (rule_tac x="rep\<cdot>x" in decisive_cases [OF d])
+apply (simp add: iso.rep_iso [OF iso])
+apply (simp add: iso.abs_strict [OF iso])
+done
+
+lemma lub_ID_finite:
+  assumes chain: "chain d"
+  assumes lub: "(\<Squnion>n. d n) = ID"
+  assumes decisive: "\<And>n. decisive (d n)"
+  shows "\<exists>n. d n\<cdot>x = x"
+proof -
+  have 1: "chain (\<lambda>n. d n\<cdot>x)" using chain by simp
+  have 2: "(\<Squnion>n. d n\<cdot>x) = x" using chain lub by (rule lub_ID_reach)
+  have "\<forall>n. d n\<cdot>x = x \<or> d n\<cdot>x = \<bottom>"
+    using decisive unfolding decisive_def by simp
+  hence "range (\<lambda>n. d n\<cdot>x) \<subseteq> {x, \<bottom>}"
+    by auto
+  hence "finite (range (\<lambda>n. d n\<cdot>x))"
+    by (rule finite_subset, simp)
+  with 1 have "finite_chain (\<lambda>n. d n\<cdot>x)"
+    by (rule finite_range_imp_finch)
+  then have "\<exists>n. (\<Squnion>n. d n\<cdot>x) = d n\<cdot>x"
+    unfolding finite_chain_def by (auto simp add: maxinch_is_thelub)
+  with 2 show "\<exists>n. d n\<cdot>x = x" by (auto elim: sym)
+qed
+
+subsection {* ML setup *}
+
 use "Tools/Domain/domain_take_proofs.ML"
 
 end